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Failure functions

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Background:  see  messages.
Abstract  definition:  Let  $\phi(x)$  be  a  function  of
$x$.  Then $x$  =  $\psi(x_0)$  is  a  failure  function  IF  $\phi(psi(x_0))$  is
a   failure  in  accordance  with  our  definition  of  a  failure. Here  $x_0$  is
a  specific  function  of  $x$.   Note  the dual  role  played  by  $x$. It  is  a  
variable  in  $\phi(x)$  and  a  function of  a  specific value  of  $x_0$  in $\psi(x_0)$.
              \line  1)  Let  our  definition  of  a  failure  be  a  composite  number.Let the
              parent  function, $\phi(x)$,    be a  polynomial ring in which  $ x$ belongs to 
              $Z$.  Then  $x  = psi(x_0)  =  x_0  +  k*\psi(x_0)$  is  a  failure function.Here
              $k$  belongs to  $Z$.